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Squares and Triangles | AIME I, 1999 | Question 4

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1999 based on Squares and Triangles.

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1999 based on Squares and triangles.

Squares and triangles – AIME I, 1999


The two squares share the same centre O and have sides of length 1, The length of AB is \(\frac{43}{99}\) and the area of octagon ABCDEFGH is \(\frac{m}{n}\) where m and n are relatively prime positive integers, find m+n.

Squares and Triangles
  • is 107
  • is 185
  • is 840
  • cannot be determined from the given information

Key Concepts


Squares

Triangles

Algebra

Check the Answer


Answer: is 185.

AIME I, 1999, Question 4

Geometry Vol I to IV by Hall and Stevens

Try with Hints


Triangle AOB, triangleBOC, triangleCOD, triangleDOE, triangleEOF, triangleFOG, triangleGOH, triangleHOA are congruent triangles

with each area =\(\frac{\frac{43}{99} \times \frac{1}{2}}{2}\)

then the area of all 8 of them is (8)\(\frac{\frac{43}{99} \times \frac{1}{2}}{2}\)=\(\frac{86}{99}\) then 86+99=185.

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